Bode Plots

The Bode plot is named after Hendrik Wade Bode (1905 – 1982), an American engineer and scientist, of Dutch ancestry, a pioneer of modern control theory and electronic telecommunications.

The frequency response function (FRF) is a complex function of the frequency $\omega=2\pi f$ that describes the response of a system to input of different frequencies:

$$H(j\omega)=\vert H(j\omega)\vert e^{j\angle H(j\omega)}=\vert H(j\omega)\vert \angle H(j\omega)$$ (494)

The Bode plot presents both the magnitude $\vert H(j\omega)\vert$ and phase angle $\angle H(j\omega)$ of $H(j\omega)$ as functions of frequency in logarithmic scale. (Zero frequency is at $-\infty$ as $10^{-\infty}=0$.) Moreover, the magnitude $\vert H(j\omega)\vert$ is also represented in logarithmic scale in decibel (dB), and is called log magnitude. A Bode plot is composed of two parts:

• The log magnitude (Lm) of $\vert H(j\omega)\vert$ with unit decibel (dB):

  $$Lm\; H(j\omega)=20 \;log_{10} \vert H(j\omega)\vert\;dB$$ (495)

• The phase plot $\angle H(j\omega)$ with either in radian or degree.

The logarithmic scale of the frequency is composed of several “decades” each for a range of frequencies from $\omega$ to $10 \omega$, independent of the specific frequency $\omega$.

Bede plots have the following advantages:

• Due to the logarithmic scale in frequency, large frequency range of several orders of magnitude can be represented;

• Convenient straight line asymptotes can be used to approximate the plots;

• The behavior of the system in terms of the magnitude, even approaching zero, can be clearly described.

• Due to the logarithmic scale of the magnitude of the FRF, multiplications and divisions of FRFs can be represented as addition and subtractions in the plot (while the phases are always added/subtracted):

  $$\left\{ \begin{array}{ll} Lm(H_1H_2)=Lm\;H_1+Lm\;H_2,&\;\;\;\;\an... ...angle H \\ Lm (1/H)=-Lm\;H,&\;\;\;\;\angle (1/H)=-\angle H \end{array} \right.$$ (496)

All FRFs of interest in this course can be considered as a combination of some components or building blocks, including:

• Constant gain $k$;
• Integral/derivative factors $(j\omega)$
• Delay factor: $e^{\pm j\omega \tau}$;
• First-order factor $(1+j\omega\tau)$;
• Second-order factor $(j\omega)^2+2\zeta\omega_n\omega j+\omega_n^2 =(\omega_n^2-\omega^2)+j\,2\zeta\omega_n$

Given the Bode plot of any building block $H(j\omega)$, we can obtain the plots of any combination of them.

We will first consider each of such components at a time, and then consider their combinations. In particular, we will study the first order system:

$$H(j\omega)=\frac{N(j\omega)}{1+j\omega \tau}$$ (497)

and the second order system:

$$H(j\omega)=\frac{N(j\omega)}{(j\omega)^2+2\zeta\omega_n j\omega +\omega_n^2} =\frac{N(j\omega)}{(\omega_n^2-\omega^2)+j\;(2\zeta\omega_n \omega) }$$ (498)